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- Composition operator (3)
- Hilbert spaces with reproducing kernels (3)
- Hardy space (2)
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- Composition operator; Adjoint; Hardy space (1)
- Composition operator; Dirichlet space (1)
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Articles 1 - 29 of 29
Full-Text Articles in Analysis
A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James Mclaughlin
A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James Mclaughlin
Mathematics Faculty Publications
We prove a generalization of Schroter's formula to a product of an arbitrary number of Jacobi triple products. It is then shown that many of the well-known identities involving Jacobi triple products (for example the Quintuple Product Identity, the Septuple Product Identity, and Winquist's Identity) all then follow as special cases of this general identity. Various other general identities, for example certain expansions of (q; q)(infinity) and (q; q)(infinity)(k), k >= 3, as combinations of Jacobi triple products, are also proved.
Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang
Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang
Mathematics Faculty Publications
In this paper, we provide a direct and constructive proof of weak factorization of h1 (ℝ×ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] in the sense of h1 (ℝ×ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ║fh1║(ℝ×ℝ) is given in terms of ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this directly implies a lower bound on the norm of …
Smirnov Class For Spaces With The Complete Pick Property, Alexandru Aleman, Michael Hartz, John E. Mccarthy, Stefan Richter
Smirnov Class For Spaces With The Complete Pick Property, Alexandru Aleman, Michael Hartz, John E. Mccarthy, Stefan Richter
Mathematics Faculty Publications
We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptanoğlu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.
Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, Ji Li, Brett D. Wick
Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, Ji Li, Brett D. Wick
Mathematics Faculty Publications
This paper provides a constructive proof of the weak factorization of the classical Hardy space in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of (the dual of ) via commutators of the multilinear Riesz transforms.
Spaces Of Dirichlet Series With The Complete Pick Property, John E. Mccarthy, Orr Moshe Shalit
Spaces Of Dirichlet Series With The Complete Pick Property, John E. Mccarthy, Orr Moshe Shalit
Mathematics Faculty Publications
We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s,u)=∑ann−s−u¯, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space Hd2 in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of Hd2. Thus, a …
The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick
The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick
Mathematics Faculty Publications
We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in C, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H and then we continue a research thread begun by Agler and McCarthy in 1999, and continued …
A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick
A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick
Mathematics Faculty Publications
Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.
Commutators In The Two-Weight Setting, Irina Holmes, Michael T. Lacey, Brett D. Wick
Commutators In The Two-Weight Setting, Irina Holmes, Michael T. Lacey, Brett D. Wick
Mathematics Faculty Publications
Let R be the vector of Riesz transforms on Rn and let μ,λ∈Ap be two weights on Rn, 1p(μ)→Lp(λ)|| is shown to be equivalent to the function b being in a BMO space adapted to μ and λ. This is a common extension of a result of Coifman–Rochberg–Weiss in the case of both λ and μ being Lebesgue measure, and Bloom in the case of dimension one.
The Von Neumann Inequality For 3 × 3 Matrices, Greg Knese
The Von Neumann Inequality For 3 × 3 Matrices, Greg Knese
Mathematics Faculty Publications
This note details how recent work of Kosiński on the three point Pick interpolation problem on the polydisc can be used to prove the von Neumann inequality for d-tuples of commuting 3 x 3 contractive matrices.
Pick Interpolation For Free Holomorphic Functions, Jim Agler, John E. Mccarthy
Pick Interpolation For Free Holomorphic Functions, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron. As an application, we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function.
Integrability And Regularity Of Rational Functions, Greg Knese
Integrability And Regularity Of Rational Functions, Greg Knese
Mathematics Faculty Publications
Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting …
Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy
Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
A Pick function of variables is a holomorphic map from to , where is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series with real numbers that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences can occur. We give an extension of Hamburger's results to Pick functions of two variables.
Comprehensive Analysis Of Escape-Cone Losses From Luminescent Waveguides, Stephen R. Mcdowall, Tristan Butler, Edward Bain, Kelsey Scharnhorst, David L. Patrick
Comprehensive Analysis Of Escape-Cone Losses From Luminescent Waveguides, Stephen R. Mcdowall, Tristan Butler, Edward Bain, Kelsey Scharnhorst, David L. Patrick
Mathematics Faculty Publications
Luminescent waveguides (LWs) occur in a wide range of applications, from solar concentrators to doped fiber amplifiers. Here we report a comprehensive analysis of escape-cone losses in LWs, which are losses associated with internal rays making an angle less than the critical angle with a waveguide surface. For applications such as luminescent solar concentrators, escape-cone losses often dominate all others. A statistical treatment of escape-cone losses is given accounting for photoselection, photon polarization, and the Fresnel relations, and the model is used to analyze light absorption and propagation in waveguides with isotropic and orientationally aligned luminophores. The results are then …
Operator Monotone Functions And Löwner Functions Of Several Variables, Jim Agler, John E. Mccarthy, N J. Young
Operator Monotone Functions And Löwner Functions Of Several Variables, Jim Agler, John E. Mccarthy, N J. Young
Mathematics Faculty Publications
We prove generalizations of Loewner's results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.
Stability Of The Gauge Equivalent Classes In Inverse Stationary Transport In Refractive Media, Stephen R. Mcdowall, Plamen Stefanov, Alexandru Tamasan
Stability Of The Gauge Equivalent Classes In Inverse Stationary Transport In Refractive Media, Stephen R. Mcdowall, Plamen Stefanov, Alexandru Tamasan
Mathematics Faculty Publications
In the inverse stationary transport problem through anisotropic attenuating, scattering, and refractive media, the albedo operator stably determines the gauge equivalent class of the attenuation and scattering coefficients.
Regression Model Fitting With Quadratic Term Leads To Different Conclusion In Economic Analysis Of Washington State Smoking Ban, Marshal Ma, Scott Mcclintock
Regression Model Fitting With Quadratic Term Leads To Different Conclusion In Economic Analysis Of Washington State Smoking Ban, Marshal Ma, Scott Mcclintock
Mathematics Faculty Publications
No abstract provided.
Gauge Equivalence In Stationary Radiative Transport Through Media With Varying Index Of Refraction, Stephen R. Mcdowall, Plamen Stefanov, Alexandru Tamasan
Gauge Equivalence In Stationary Radiative Transport Through Media With Varying Index Of Refraction, Stephen R. Mcdowall, Plamen Stefanov, Alexandru Tamasan
Mathematics Faculty Publications
Three dimensional anisotropic attenuating and scattering media sharing the same albedo operator have been shown to be related via a gauge transformation. Such transformations define an equivalence relation. We show that the gauge equivalence is also valid in media with non-constant index of refraction, modeled by a Riemannian metric. The two dimensional model is also investigated.
Optical Tomography For Media With Variable Index Of Refraction, Stephen R. Mcdowall
Optical Tomography For Media With Variable Index Of Refraction, Stephen R. Mcdowall
Mathematics Faculty Publications
Optical tomography is the use of near-infrared light to determine the optical absorption and scattering properties of a medium M ⊂ Rn. If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric and photons which do not get absorbed or scatter travel along straight lines. In this expository article we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics …
Sublimital Analysis, Thomas Q. Sibley
Sublimital Analysis, Thomas Q. Sibley
Mathematics Faculty Publications
The Bolzano-Weierstrass theorem asserts, under appropriate circumstances, the convergence of some subsequence of a sequence. While this famous theorem ignores the actual limit of the subsequence, it is natural to investigate such limits. This note characterizes the set of possible limits of subsequences of a given sequence.
Adjoints Of Composition Operators With Rational Symbol, Christopher Hammond, Jennifer Moorhouse, Marian Robbins
Adjoints Of Composition Operators With Rational Symbol, Christopher Hammond, Jennifer Moorhouse, Marian Robbins
Mathematics Faculty Publications
Building on techniques developed by C. C. Cowen and E. A. Gallardo-Gutiérrez [J. Funct. Anal. 238 (2006), no. 2, 447–462;MR2253727 (2007e:47033)], we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space H 2 . We consider some specific examples, comparing our formula with several results that were previously known.
Composition Operators With Maximal Norm On Weighted Bergman Spaces, Brent J. Carswell, Christopher Hammond
Composition Operators With Maximal Norm On Weighted Bergman Spaces, Brent J. Carswell, Christopher Hammond
Mathematics Faculty Publications
We prove that any composition operator with maximal norm on one of the weighted Bergman spaces is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space H2, where every inner function induces a composition operator with maximal norm.
Isolation And Component Structure In Spaces Of Composition Operators, Christopher Hammond, Barbara D. Maccluer
Isolation And Component Structure In Spaces Of Composition Operators, Christopher Hammond, Barbara D. Maccluer
Mathematics Faculty Publications
We establish a condition that guarantees isolation in the space of composition operators acting between H p (B N ) and H q (B N ), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.
The Norm Of A Composition Operator With Linear Symbol Acting On The Dirichlet Space, Christopher Hammond
The Norm Of A Composition Operator With Linear Symbol Acting On The Dirichlet Space, Christopher Hammond
Mathematics Faculty Publications
We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.
Norms Of Linear-Fractional Composition Operators, Paul S. Bourdon, E. E. Fry, Christopher Hammond, C. H. Spofford
Norms Of Linear-Fractional Composition Operators, Paul S. Bourdon, E. E. Fry, Christopher Hammond, C. H. Spofford
Mathematics Faculty Publications
No abstract provided.
An Inverse Problem For The Transport Equation In The Presence Of A Riemannian Metric, Stephen R. Mcdowall
An Inverse Problem For The Transport Equation In The Presence Of A Riemannian Metric, Stephen R. Mcdowall
Mathematics Faculty Publications
The stationary linear transport equation models the scattering and absorption of a low-density beam of neutrons as it passes through a body. In Euclidean space, to a first approximation, particles travel in straight lines. Here we study the analogous transport equation for particles in an ambient field described by a Riemannian metric where, again to first approximation, particles follow geodesics of the metric. We consider the problem of determining the scattering and absorption coefficients from knowledge of the albedo operator on the boundary of the domain. Under certain restrictions, the albedo operator is shown to determine the geodesic ray transform …
Fixed Points Of Holomorphic Mappings For Domains In Banach Spaces, Lawrence A. Harris
Fixed Points Of Holomorphic Mappings For Domains In Banach Spaces, Lawrence A. Harris
Mathematics Faculty Publications
We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions.
A Construction Of Compactly-Supported Biorthogonal Scaling Vectors And Multiwavelets On $R^2$, Bruce Kessler
A Construction Of Compactly-Supported Biorthogonal Scaling Vectors And Multiwavelets On $R^2$, Bruce Kessler
Mathematics Faculty Publications
In \cite{K}, a construction was given for a class of orthogonal compactly-supported scaling vectors on $\R^{2}$, called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continuous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continuous piecewise linear functions on a uniform subdivision of the …
An Electromagnetic Inverse Problem In Chiral Media, Stephen R. Mcdowall
An Electromagnetic Inverse Problem In Chiral Media, Stephen R. Mcdowall
Mathematics Faculty Publications
We consider the inverse boundary value problem for Maxwell's equations that takes into account the chirality of a body in R3 . More precisely, we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters, namely the conductivity, electric permittivity, magnetic permeability and chirality, in the interior. We rewrite Maxwell's equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics.
Total Determination Of Material Parameters From Electromagnetic Boundary Information, M. S. (Mark Suresh) Joshi, Stephen R. Mcdowall
Total Determination Of Material Parameters From Electromagnetic Boundary Information, M. S. (Mark Suresh) Joshi, Stephen R. Mcdowall
Mathematics Faculty Publications
In this paper we complete the proof that the material parameters can be obtained for a chiral electromagnetic body from the boundary admittance map. We prove that from the admittance map, the parameters are uniquely determined to infinite order at the boundary. This removes the assumption of such knowledge in the result of the second author regarding interior determination for chiral media.